Problem: Let $y=\sec(x)$. What is the value of $\dfrac{dy}{dx}$ at $x=\dfrac{\pi}{4}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac12$ (Choice B) B $\dfrac{1}{\sqrt{2}}$ (Choice C) C $\sqrt2$ (Choice D) D $1$
Solution: Let's first find $\dfrac{dy}{dx}$. Then, we can evaluate it at $x=\dfrac{\pi}{4}$. Recall that the derivative of $\sec(x)$ is $\dfrac{\sin(x)}{\cos^2(x)}$, or $\sec(x)\tan(x)$. Put another way, $\dfrac{d}{dx}[\sec(x)]=\dfrac{\sin(x)}{\cos^2(x)}=\sec(x)\tan(x)$. [Is there a way to know this without memorizing?] Now let's plug in $x={\dfrac{\pi}{4}}$ : $\begin{aligned} &\phantom{=}\dfrac{\sin\left({\dfrac{\pi}{4}}\right)}{\cos^2\left({\dfrac{\pi}{4}}\right)} \\\\ &=\dfrac{\dfrac{\sqrt2}{2}}{\left(\dfrac{\sqrt{2}}{2}\right)^2} \\\\ &={\dfrac{\sqrt2}2}\cdot\dfrac{4}{2} \\\\ &=\sqrt2 \end{aligned}$ In conclusion, the value of $\dfrac{dy}{dx}$ at $x=\dfrac{\pi}{4}$ is $\sqrt2$.